Apparently the free version of Hamachi requires the GUI to be running in order to operate. When you disconnect your RDP session the free version of Hamachi will disconnect. The workaround appears to be to run the following command instead of disconnecting. (Note: Number 1 below is your session ID. To know what your current session ID, in the console, run
query user or open the task manager and go to users and look at your session ID)
tscon 1 /dest:console
This may need to be run as administrator. This will transfer your session back to the local machine. Also note that when the session is transfered back to the local machine it will not be locked and will be available to anyone with physical access to the machine.